Equality is a fundamental concept in mathematics. The property of equality can be generalized to the concept of sameness, which allows us to talk about statements that are true for all elements of a set.
There are three properties associated with equality: reflexivity, symmetry, and transitivity. These properties are used to prove that a statement is true for all elements of a set.
Reflexivity is the property that states that every element in a set is equal to itself. For example, the set of all even numbers is equal to itself because every even number is equal to itself.
Symmetry is the property that states that two elements in a set are equal if and only if their images are also equal. For example, if two numbers are divisible by two (that is, they have a remainder of zero when divided by two), then they are also divisible by four and eight.
Transitivity is the property that states that if one element in a set is equal to another and another element in the same set is also equal to it, then all three elements are equal to each other. For example, if 4=3=1+3 and 5=4+1+4+1+2=2+2+2+2+2 then 3=2+2+2+2 (or equivalently 1=1).